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Complete solution to open problem on exponential arithmetic-geometric index

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  • Mondal, Sourav
  • Raza, Zahid

Abstract

One of the well-studied topics in extremal graph theory is the identification of extremal unicyclic graphs with respect to topological indices. Cruz, Rada, and Sanchez [MATCH Commun. Math. Comput. Chem. 88 (2022) 481–503] proposed a unified framework for identifying extremal unicyclic graphs with respect to degree-based topological indices, given the graph order. The exponential arithmetic-geometric index (EAG) is a degree-based index, which is defined for a graph G asEAG(G)=∑vivj∈E(G)edG(vi)+dG(vj)2dG(vi)dG(vj),where dG(vi) represents the degree of the vertex vi, and E(G) denotes the graph’s edge set. The EAG index was excluded from the aforementioned framework due to its unique counting characteristics. Consequently, determining maximal unicyclic graphs for EAG was identified as an open problem in the same study. This work aims to address this gap by providing a comprehensive solution. We identify the maximal unicyclic graph for EAG with given graph order n. In addition, we explore sharp upper and lower bounds of EAG for trees as functions of graph order. The extremal chemical trees are also generated. Furthermore, we report regression relationships between EAG and the physicochemical properties of octanes.

Suggested Citation

  • Mondal, Sourav & Raza, Zahid, 2026. "Complete solution to open problem on exponential arithmetic-geometric index," Applied Mathematics and Computation, Elsevier, vol. 513(C).
    • Handle: RePEc:eee:apmaco:v:513:y:2026:i:c:s009630032500534x
    DOI: 10.1016/j.amc.2025.129809
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    References listed on IDEAS

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    1. Bermudo, Sergio & Cruz, Roberto & Rada, Juan, 2022. "Vertex-degree-based topological indices of oriented trees," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    2. Mondal, Sourav & Das, Kinkar Chandra, 2024. "Complete solution to open problems on exponential augmented Zagreb index of chemical trees," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    3. Cruz, Roberto & Monsalve, Juan & Rada, Juan, 2020. "Extremal values of vertex-degree-based topological indices of chemical trees," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    4. Vujošević, Saša & Popivoda, Goran & Kovijanić Vukićević, Žana & Furtula, Boris & Škrekovski, Riste, 2021. "Arithmetic–geometric index and its relations with geometric–arithmetic index," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    5. Roberto Cruz & Juan Daniel Monsalve & Juan Rada, 2021. "The balanced double star has maximum exponential second Zagreb index," Journal of Combinatorial Optimization, Springer, vol. 41(2), pages 544-552, February.
    6. Das, Kinkar Chandra, 2024. "Open problems on Sombor index of unicyclic and bicyclic graphs," Applied Mathematics and Computation, Elsevier, vol. 473(C).
    7. Rada, Juan, 2017. "Vertex-degree-based topological indices of hexagonal systems with equal number of edges," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 270-276.
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